A078923 Possible values of sigma(n)-n.

0, 1, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74

Offset: 1

Author

Benoit Cloitre, Dec 15 2002

Keywords

• nonn

Comments

To test whether k>1 is in the sequence, it suffices to check values of n up to (k-1)^2, since sigma(n)-n >= sqrt(n)+1 if n is composite.

Erdős (Elem. Math. 28 (1973), 83-86) shows that the density of even integers in the range of a(n) is strictly less than 1/2. The argument of Coppersmith (1987) shows that the range of a(n) has density at most 47/48 < 1. - N. J. A. Sloane, Dec 21 2019

The lower asymptotic density is at least 1/2 by the 'almost all' binary Goldbach conjecture, independently proved by Nikolai Chudakov, Johannes van der Corput, and Theodor Estermann. (In this context, this shows that the density of the odd numbers of this form is 1 (consider A001065(p*q) for prime p, q); full Goldbach would prove that 5 is the only odd number absent from this sequence.) - Charles R Greathouse IV, Dec 14 2022

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)
• Don Coppersmith, An answer to the problem of Don Saari, 1987.
• Paul Erdős, Über die Zahlen der Form sigma(n)-n und n-phi(n), Elemente der Math., Vol. 28 (1973), pp. 83-86; alternative link.

Crossrefs

Cf. A000203, A001065, A002191, A007369. Complement of A005114.

Programs

PARI

lista(nn)=for (n=0, nn, if (n==1, kmax=2, kmax=(n-1)^2); for (k=1, kmax, if (sigma(k)-k == n, print1(n, ", "); break););); \\ Michel Marcus, Nov 11 2014

Extensions

Edited by Dean Hickerson, Dec 19 2002

Offset fixed by Michel Marcus, Dec 19 2014